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What exactly is an adjoint flux?

  1. Jul 6, 2010 #1
    I've been reading a bit about hybrid methods, and I keep coming across adjoint fluxes. What exactly is an adjoint flux? And how does it factor in to calculations?


  2. jcsd
  3. Jul 6, 2010 #2


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    An adjoint calculation is basically a backwards calculation. Instead of starting with a neutron and calculating where it goes, you start with a fission event and calculate where the neutron that caused it could have come from.

    Adjoint flux is basically an importance factor for neutrons, and it's useful for radiation shielding and criticality analysis.
  4. Jul 10, 2010 #3


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    Mathematical operators have "adjoints". For example, the adjoint of a real matrix is its
    transpose. The adjoint of the derivative is its negative.

    You can take the Boltzmann transport equation, and take the adjoint of each of its
    terms, and you get the adjoint Boltzmann equation. The solution to this equation is
    the adjoint flux.

    There is a physical interpretation to the adjoint equation; the concept of neutron
    "importance" obeys the adjoint equation.

    When you define certain quantities like reactivity, you do so with a weighting function
    in general. If you use the adjoint flux as your weighting function, then your calculation
    of the quantity becomes accurate to second order instead of first order when you use
    first order perturbation theory.

    Let H be the Boltzmann transport equation operator, and H* is its adjoint.

    We have then psi as the solution to the forward equation - H psi = 0
    We also have (H*)(psi*) = 0.

    Suppose H' = H + dH and psi' = psi + dpsi

    <psi*| H' | psi'> = <psi*| H + dH | psi + dpsi > =
    <psi*| H | psi > + <psi*| H | dpsi> + <psi*| dH | psi > + <psi*| dH | dpsi >

    The first term is zero [ H psi = 0 ]. First order perturbation theory would have one
    using the third term as the answer. But the second term which we can't evaluate without
    solving the new system for dpsi is also first order. However, from the definition of adjoints;

    <psi*|H|dpsi> = < psi* | H dpsi > = < H*psi*|dpsi> = 0 since H* psi* = 0

    Therefore, if you use only the 3rd term as per first order perturbation theory - your
    error - the 4 th term is 2nd order - it has a dH and a dpsi.

    The other first order term vanishes if you weight with the adjoint.

    Dr. Gregory Greenman
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